Geometry
# 3D Coordinate Geometry

The points \(A\), \(B\), \(P\), and \(Q\) all lie on one line, with \(A=(-3, 5, 8).\)

Point \(P\) lies in the \(xy\)-plane between \(A\) and \(B\) such that the distance from \(A\) to \(P\) is twice the distance from \(P\) to \(B\). Furthermore, \(Q\) sits on the \(z\)-axis such that the distance from \(A\) to \(Q\) is twice the distance from \(Q\) to \(B\).

If \(B=(x,y,z),\) what is the value of \(x+y+z?\)

An infinite column is centered along the \(z\)-axis. It has a square cross-section of side length 10. It is cut by the plane \(4x - 7y + 4z = 25.\)

What is the area of the surface cut?

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